Related papers: On divisibility concerning binomial coefficients

It is well-known that for any distinct positive integers $k$ and $n$, the numbers $2^{2^k}+1$ and $2^{2^n}+1$ are relatively prime. In this paper we consider the situation when 1 is replaced by some positive integer $d>1$

We know [Rui Duarte and Ant\'onio Guedes de Oliveira, New developments of an old identity, manuscript arXiv:1203.5424, submitted.] that, for every non-negative integer numbers $n,i,j$ and for every real number $\ell$, $$ \sum_{i+j=n}…

We give a new proof of the following statement: the Catalan number $C_n$ is divisible by $n+2$, if $n$ is odd and $n\not\equiv 1\text{ mod }3$.

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack…

Let $n\geq2$ be an integer. In this paper, we study the convexity of the so-called MacMahon's $q$-Catalan polynomials $C_n(q)=\frac1{[n+1]_q}\left[ 2n \atop n \right]_q$ as functions of $q$. Along the way, several intermediate results on…

In his book "250 Problems in Elementary Number Theory", W.Sierpinski shows that the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... In this paper, we prove a similar but more general result.Consider the natural numbers of…

We prove exact asymptotic expansions for the partial sums of the sequences of central binomial coefficients and Catalan numbers, $\sum_{k=0}^n \binom{2k}{k}$ and $\sum_{k=0}^n C_n$. We also obtain closed forms for the polynomials…

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for…

Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p.…

We present yet another algebraic proof of the unimodality of the binomial coefficients.

Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of…

Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order…

A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting…

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations.…

We determine all triples $(a,b,n)$ of positive integers such that $a$ and $b$ are relatively prime and $n^k$ divides $a^n + b^n$ (respectively, $a^n - b^n$), when $k$ is the maximum of $a$ and $b$ (in fact, we answer a slightly more general…

Suppose $k,x,$ and $b$ are positive integers, and $a$ is a nonnegative integer such that $k=a+b$. In this paper, we will prove $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ if and only if $x=a=1$. We do this by looking at different cases…

Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial…

In this work, we completely characterize (i) permutation binomials of the form $x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb{F}_{2^n}[x], n = 2^st, a \in \mathbb{F}_{2^{2t}}^{*}$, and (ii) permutation trinomials of the form…

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For…

Binomial Theorem for (N+n)^r is described with non-commuting variables N and n.